A recent question about combinatorial interpretations, namely to find an interpretation for the identity
$$\sum_{k=0}^m 2^{-2k} \binom{2k}{k} \binom{2m-k}{m}=4^{-m} \binom{4m+1}{2m}$$
gives another full class of such issues. This equation is much more than a tautology when interpreted intensionally as being about combinatorics: a combinatorial interpretation involves give a natural class $C_l$ of objects for the left-hand-side and $C_r$ for the right-hand side and a bijection between these classes. Furthermore, and this is where things get really interesting (with respect to my original question), such classes $C_l$ and $C_r$ would be considered 'natural' if by using the usual rules of combinatorial counting, we would naturally get that the number of objects of $C_l$ of size $m$ is the left-hand side expression (similarly for $C_r$ and the rhs). The point is that these counting expressions would be derived structurally from the combinatorial classes. This is a much more interesting interplay between 3 extensional objects (a counting function and 2 combinatorial classes) and 2 intensional ones (different-but-equal formulas representing the counting function).
